350 



PROCEEDINGS OF THE AMERICAN ACADEMY 



We must now find V. Let AB he the coil bounding the magnetic 

 shell, and let CZ be the axis of the coil, and 

 consequently normal to the shell. Let us take 

 a point Z upon the axis, distant z from the 

 centre C. The potential at Z due to any small 

 element d S at P upon the surface of the mag- 

 netic shell is 



dS 



ZP 



but 



ZP =z (2^ J^a^—2az cosa)^ 



calling cos a = /*, 



also, 



ZP zV ^2 z^y 



Yp = \i}^t-^lr 



calling 



h and - = h^, we have 



l- = \{l-2ht^^h')-\ 



ZP 



^=ki-2^i/^ + vr^- 



ZP 



[Equa. I.] 

 [Equa. II.] 



Developing equation [I.] by the binomial theorem, we have 



-ip = l[l+MA + (-'^V^-i)^^+G;^«-|^)A^ + &c.] [III.] 



= - {Po+P.h + P.,h' + PJ^' + &c.). 



[IV.] 



Equation IV. is merely a simplified form of equation III., the expres- 

 sions Pp, Pj, P2, P„ &c. being substituted for the coefficients of A" or 1, 

 h, h\ h', &c. In the same way equation II. becomes 



-L^ = 1 (Po + p,A, + p, V + PzK + &c.). [V] 



